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A laser beam is called diffraction-limited if its potential to be focused to small spots is as high as possible for the given wavelength, i.e., if its beam quality is ideal.
More precisely, this means that a beam waist with a given beam radius, generated from the beam by focusing e.g. with a curved mirror, is associated with the minimum possible beam divergence.
What exactly this means depends on the definition of the beam radius and beam divergence.
If second moments are used for the definition of both quantities, the minimum beam parameter product is reached for a Gaussian beam, which has not only a Gaussian intensity profile but also a flat phase front at the beam waist.

A laser which operates on a single transverse resonator mode will usually have a diffraction-limited output, since that single mode is usually the fundamental resonator mode, and the shape of this is usually close to Gaussian.
Non-ideal beam quality can be caused by intracavity beam distortions, e.g. in the gain medium, which can either deform the fundamental mode or (more frequently) cause transverse multimode operation.
Note that the beam quality of a laser depends not only on the strength of intracavity distortions, but also on certain details of the resonator design; optimization of the latter can be essential for reaching a diffraction-limited output.

Arbitrary Index Profiles

A fiber's index profile may be more complicated than just a circle:

Here, we "printed" some letters, translated this into an index profile and initial optical field, propagated the light over some distance and plotted the output field – all automated with a little script code.